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Theorem cntz2ss 17759
Description: Centralizers reverse the subset relation. (Contributed by Mario Carneiro, 3-Oct-2015.)
Hypotheses
Ref Expression
cntzrec.b 𝐵 = (Base‘𝑀)
cntzrec.z 𝑍 = (Cntz‘𝑀)
Assertion
Ref Expression
cntz2ss ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))

Proof of Theorem cntz2ss
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2621 . . . . . 6 (+g𝑀) = (+g𝑀)
2 cntzrec.z . . . . . 6 𝑍 = (Cntz‘𝑀)
31, 2cntzi 17756 . . . . 5 ((𝑥 ∈ (𝑍𝑆) ∧ 𝑦𝑆) → (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
43ralrimiva 2965 . . . 4 (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
5 ssralv 3664 . . . . 5 (𝑇𝑆 → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
65adantl 482 . . . 4 ((𝑆𝐵𝑇𝑆) → (∀𝑦𝑆 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
74, 6syl5 34 . . 3 ((𝑆𝐵𝑇𝑆) → (𝑥 ∈ (𝑍𝑆) → ∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
87ralrimiv 2964 . 2 ((𝑆𝐵𝑇𝑆) → ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥))
9 cntzrec.b . . . 4 𝐵 = (Base‘𝑀)
109, 2cntzssv 17755 . . 3 (𝑍𝑆) ⊆ 𝐵
11 sstr 3609 . . . 4 ((𝑇𝑆𝑆𝐵) → 𝑇𝐵)
1211ancoms 469 . . 3 ((𝑆𝐵𝑇𝑆) → 𝑇𝐵)
139, 1, 2sscntz 17753 . . 3 (((𝑍𝑆) ⊆ 𝐵𝑇𝐵) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
1410, 12, 13sylancr 695 . 2 ((𝑆𝐵𝑇𝑆) → ((𝑍𝑆) ⊆ (𝑍𝑇) ↔ ∀𝑥 ∈ (𝑍𝑆)∀𝑦𝑇 (𝑥(+g𝑀)𝑦) = (𝑦(+g𝑀)𝑥)))
158, 14mpbird 247 1 ((𝑆𝐵𝑇𝑆) → (𝑍𝑆) ⊆ (𝑍𝑇))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  wral 2911  wss 3572  cfv 5886  (class class class)co 6647  Basecbs 15851  +gcplusg 15935  Cntzccntz 17742
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-ov 6650  df-cntz 17744
This theorem is referenced by:  cntzidss  17764  gsumzadd  18316  dprdfadd  18413  dprdss  18422  dprd2da  18435  dmdprdsplit2lem  18438  cntzsdrg  37598
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